The Helmholtz Theorem

When the divergence and the curl $D(r),C(r)$ of a vector function $F(r)$ are known and go to zero faster than $\frac{1}{r^2}$ as $r\to \infty$, then there exists a unique solution, $F$ to

$$F=-\nabla U+\nabla \times W$$

Therefore, $F$ is uniquely given by a combination of curl and divergence of two functions $U,W$ containing $D(r),C(r)$, respectively.

$$U(r)\equiv \frac{1}{4\pi }\int \frac{D(r^{\prime })}{r}d{\tau }^{\prime }$$ $$W(r)\equiv \frac{1}{4\pi }\int \frac{C(r^{\prime })}{r}d{\tau }^{\prime }$$